Hidden Markov Models

1. Hidden Markov Models Dave DeBarr ddebarr@gmu.edu

2. Overview • General Characteristics • Simple Example • Speech Recognition

3. Andrei Markov • Russian statistician (1856 – 1922) • Studied temporal probability models • Markov assumption • Statet depends only on a bounded subset of State0:t-1 • First-order Markov process • P(Statet | State0:t-1) = P(Statet | Statet-1) • Second-order Markov process • P(Statet | State0:t-1) = P(Statet | Statet-2:t-1)

4. Hidden Markov Model (HMM) • Evidence can be observed, but the state is hidden • Three components • Priors (initial state probabilities) • State transition model • Evidence observation model • Changes are assumed to be caused by a stationary process • The transition and observation models do not change

5. Simple HMM • Security guard resides in underground facility (with no way to see if it is raining) • Wants to determine the probability of rain given whether the director brings an umbrella • P(Rain0 = t) = 0.50

6. What can you do with an HMM? • Filtering • P(Statet | Evidence1:t) • Prediction • P(Statet+k | Evidence1:t) • Smoothing • P(Statek | Evidence1:t) • Most likely explanation • argmaxState1:t P(State1:t | Evidence1:t)

7. Filtering(the forward algorithm) P(Rain1 = t) = ΣRain0 P(Rain1 = t | Rain0) P(Rain0) =0.70 * 0.50 + 0.30 * 0.50 = 0.50 P(Rain1 = t | Umbrella1 = t) = α P(Umbrella1 = t | Rain1 = t) P(Rain1 = t) = α * 0.90 * 0.50 = α *0.45 ≈ 0.818 P(Rain2 = t | Umbrella1 = t) = ΣRain1 P(Rain2 = t | Rain1) P(Rain1 | Umbrella1 = t) = 0.70 * 0.818 + 0.30 * 0.182 ≈ 0.627 P(Rain2 = t | Umbrella1 = t, Umbrella2 = t) = α P(Umbrella2 = t | Rain2 = t) P(Rain2 = t | Umbrella1 = t) = α * 0.90 * 0.627 ≈ α * 0.564 ≈ 0.883

8. Smoothing(the forward-backward algorithm) P(Umbrella2 = t | Rain1 = t) = ΣRain2 P(Umbrella2 = t | Rain2) P(* | Rain2) P(Rain2 | Rain1 = t) = 0.9 * 1.0 * 0.7 + 0.2 * 1.0 * 0.3 = 0.69 P(Rain1 = t | Umbrella1 = t, Umbrella2 = t) = α * 0.818 * 0.69 ≈ α * 0.56 ≈ 0.883

9. Most Likely Explanation(the Viterbi algorithm) P(Rain1 = t, Rain2 = t | Umbrella1 = t, Umbrella2 = t) = P(Umbrella1 = t | Rain1 = t) * P(Rain2 = t | Rain1 = t) * P (Umbrella2 = t | Rain2 = t) = 0.818 * 0.70 * 0.90 ≈ 0.515

10. Speech Recognition(signal preprocessing)

11. Speech Recognition(models) • P(Words | Signal) = α P(Signal | Words) P(Words) • Decomposes into an acoustic model and a language model • Ceiling or Sealing • High ceiling or High sealing • A state in a continuous speech HMM may be labeled with a phone, a phone state, and a word

12. Speech Recognition(phones) • Human languages use a limited repertoire of sounds

13. Acoustic signal for [t] Silent beginning Small explosion in the middle (Usually) Hissing at the end Speech Recognition(phone model)

14. Speech Recognition(pronounciation model) • Coarticulation and dialect variations

15. Speech Recognition(language model) • Can be as simple as bigrams P(Wordi | Word1:i-1) = P(Wordi | Wordi-1)

16. References • Artificial Intelligence: A Modern Approach • Second Edition (2003) • Stuart Russell & Peter Norvig • Hidden Markov Model Toolkit (HTK) • http://htk.eng.cam.ac.uk/ • Nice tutorial (from data prep to evaluation)